Conference on Computational Physics

Conference on Computational Physics - 2008
LYAPUNOV EXPONENTS OF THE GENERALIZED LOGISTIC
MAP
B. C. T. Cabella† , A. L. Espı́ndola, A. S. Martinez
Departamento de Fı́sica e Matemática, Faculdade de Filosofia, Ciências e
Letras de Ribeirão Preto, Universidade de São Paulo.
E-mail: [email protected]
Recently, the generalization of the logarithmic and exponential functions has
attracted the attention of researches. One-parameter generalization of logarithmic and exponential functions have been proposed [1, 2, 3, 4, 5]. A particular one-parameter generalization of the logarithm and exponential function,
based only in geometric arguments coincide with the ones of non-extensive
statistical mechanics. They are suitable to unify the most of the popular
one-specie continuous and discrete population dynamics models into simple
formulas [6, 7, 8, 9, 10]. In this work we discretize the Richards’ model that is
suitably written in terms of the generalized logarithm function and generalizes the Verhulst and Gompetz models. From this discretization, one obtains
a generalized logistic map. This map presents a route to chaos similar to
the logistic map but with different Lyapunov constants. We present the numerical calculation of these constants and discuss the universality class this
generalized logistic maps that it belongs.
[1] C. Tsallis, J. Stat. Phys. 52, 479 (1998).
[2] C. Tsallis, Quim. Nova 17, 468 (1994).
[3] L. Nivanem et al., Rep. Math. Phys. 52, 437 (2003).
[4] E. P. Borges, Physica A 340, 95 (2004).
[5] N. Kalogeropoulos, Physica A 356, 408 (2005).
[6] C. Tsallis et al., Physica A 233, 395 (1996).
[7] D. O. Cajueiro, Physica A 364, 385 (2006).
[8] D. O. Cajueiro et al., Physica A 373, 593 (2007).
[9] C. Anteneodo et al., Europhys. Lett. 59, 635 (2002).
[10] A. de Jesus Holanda et al., Physica A 344, 530 (2004).